Nous pouvons entrer dans l’immeuble avec le badge.

Yes. Pouvons is the nous form of the irregular verb pouvoir in the present tense.

The present-tense forms are:

• je peux
• tu peux
• il / elle / on peut
• nous pouvons
• vous pouvez
• ils / elles peuvent

So with nous, you must use pouvons.

A very common spoken alternative is on peut, which often replaces nous pouvons in everyday conversation. Nous pouvons sounds a bit more formal or written, but it is completely normal French.

It can cover all of those ideas, depending on context. French pouvoir is broader than one single English word.

In this sentence, it most naturally suggests:

• ability/access: we are able to get in
• possibly also permission/authorization: we are allowed to get in

Because of avec le badge, the idea is probably that the badge gives us access.

Entrer by itself can mean to go in / to enter when the place is already understood:

• Nous pouvons entrer.

But when you want to name the place, French very often uses entrer dans + place:

• entrer dans la maison
• entrer dans la salle
• entrer dans l’immeuble

So dans shows movement into the inside of the building.

Entrer l’immeuble would not mean enter the building in normal French.

Because dans fits the idea of going into something.

• dans = into / inside
• à often points more to a location, like at or to

Since the sentence is about entering the inside of the building, dans is the natural choice.

So:

• entrer dans l’immeuble = go into the building

Using à here would not sound natural for this meaning.

Because French uses elision before a vowel sound.

The article le becomes l’ before a word beginning with a vowel or silent h:

• le livre
• l’immeuble
• l’hôtel

So le immeuble becomes l’immeuble.

You actually cannot tell from l’ alone.

Both le and la become l’ before a vowel:

• le immeuble → l’immeuble
• la école → l’école

So when you see l’, the gender is hidden. You have to know it from a dictionary, from another sentence, or from prior learning.

In this case, immeuble is masculine:

• un immeuble
• le immeuble → l’immeuble

Le usually points to something specific and identifiable.

Here, le badge probably means a particular badge already known in the situation, such as:

• the building access badge
• the badge we have been talking about
• the badge assigned to us

If you said un badge, it would sound more indefinite, like a badge / some badge, not a specific one already understood.

Literally, it means with the badge, but in this sentence it most naturally means using the badge or by means of the badge.

So avec here expresses the tool or means that makes the action possible.

It does not only mean physical accompaniment. It is not just we are carrying the badge with us. The more natural reading is:

• the badge is what allows entry

If you wanted to be more explicit, French could also use expressions like grâce au badge or à l’aide du badge, but avec le badge is simple and natural.

Yes. French word order is somewhat flexible here.

For example, you could also say:

• Avec le badge, nous pouvons entrer dans l’immeuble.

That version puts extra emphasis on the badge.

The original order is very natural because French often puts this kind of detail after the main action:

• Nous pouvons entrer dans l’immeuble avec le badge.

So both are possible, but the original sentence is a very standard way to say it.

A rough English-style guide is:

Noo poo-von ahn-tray dahn lee-muhbl ah-vek luh bahj

A few important points:

• nous: the s is silent
• pouvons: the final s is silent
• entrer ends with an ay sound
• dans has a nasal vowel; the final s is silent
• l’immeuble begins with an l sound because of l’
• badge is pronounced like bahj

One useful detail: in normal speech, you do not usually pronounce a linking z between pouvons and entrer here.